(a) In Problem 15, explain why it is sufficient to analyze only (b) Suppose k1...

(a) In Problem 15, explain why it is sufficient to analyze only

(b) Suppose k₁ = 0.2, k₂ = 0.7, and n = 10. Choose various values of i(0) = i0, 0 < i₀ < 10. Use a numerical solver to determine what the model predicts about the epidemic in the two cases s0 > k₂/k₁ and In the case of an epidemic, estimate the number of people who are eventually infected.

(reference problem 15)

SIR Model A communicable disease is spread throughout a small community, with a fixed population of n people, by contact between infected individuals and people who are susceptible to the disease. Suppose that everyone is initially susceptible to the disease and that no one leaves the community while the epidemic is spreading. At time t, let s(t), i(t), and r(t) denote, in turn, the number of people in the community (measured in hundreds) who are susceptible to the disease but not yet infected with it, the number of people who are infected with the disease, and the number of people who have recovered from the disease. Explain why the system of differential equations

where k₁ (called the infection rate) and k₂ (called the removal rate) are positive constants, is a reasonable mathematical model, commonly called a SIR model, for the spread of the epidemic throughout the community. Give plausible initial conditions associated with this system of equations.